After you've used the elevator to get to the upper level of the Soldier's Field find a ticket pavilion building (TICKETS sign). There's a cannon in this building and a Vox code book inside its barrel. Collecting this item will start the mission.
Across the ticket pavilion is a Fellow Traveler bar. Locate the bathrooms behind the bar. One of the bathrooms hides a Vox code painted on a wall. Assuming you already have the code book you only have to enter this bathroom and Elizabeth will read the code.
In order to get to the Vox secret room you must go to the other bathroom. Locate a hat hanger and interact with it. As a result one of the walls will move, unlocking a narrow passageway.
The secret room contains weapons, ammunition, medical supplies and money. There's a small table in the middle of the room. You can find three valuable items here - a lockpick, Infusion #7 and Voxophone #29: Their Sun is Setting - Daisy Fitzroy. Once you've searched the room leave it and resume your primary mission.